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  • The tangents to a parabola at the vertex V and any point P meet at Q. If S be the focus then SP, SQ, SV are in     <div class='qna-

The tangents to a parabola at the vertex V and any point P meet at Q. If S be the focus then SP, SQ, SV are in 

 

 

Option: 1

A.P.  


Option: 2

G.P.


Option: 3

H.P.


Option: 4

 none of these


Answers (1)

best_answer

Let the parabola be \mathrm{y^2=4 a x}.
Q is the intersection of the lines \mathrm{x=0} and  \mathrm{t y=x+a t^2}, where \mathrm{P \equiv\left(a t^2, 2 a t\right)}.

Solving these, \mathrm{Q=(0, a t)}. Also \mathrm{S=(a, 0)}.

\begin{array}{ll} \therefore & \mathrm{S P^2=\left\{a\left(t^2-1\right)\right\}^2+4 a^2 t^2=a^2\left(t^2+1\right)^2} \\ & \mathrm{S Q^2=a^2+a^2 t^2=a^2\left(t^2\right)+1 \text { and } S V=a }\\ \therefore &\mathrm{ S Q^2=S P . S V} \end{array}

Posted by

Rishabh

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